Subgroups — #30664172

Order 16768.

38 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃₁⋊C₆₄	C₂	1	2	yes	maximal, complemented
2	C₂×C₃₂×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×C₁₃₁	C₄	1	1	yes	characteristic
4	C₃₂×C₁₃₁	C₄	1	1	yes	characteristic
5	C₃₂×C₁₃₁	C₂²	1	1	yes	characteristic
6	C₂×C₈×C₁₃₁	C₈	1	1	yes	characteristic
7	C₁₆×C₁₃₁	C₈	1	1	yes	characteristic
8	C₁₆×C₁₃₁	C₂×C₄	1	1	yes	characteristic
9	C₂×C₄×C₁₃₁	C₁₆	1	1	yes	characteristic
10	C₈×C₁₃₁	C₁₆	1	1	yes	characteristic
11	C₈×C₁₃₁	C₂×C₈	1	1	yes	characteristic
12	C₂²×C₁₃₁	C₃₂	1	1	yes	characteristic
13	C₄×C₁₃₁	C₃₂	1	1	yes	characteristic
14	C₄×C₁₃₁	C₂×C₁₆	1	1	yes	characteristic
15	C₂×C₁₃₁	—	2	1	no	—
16	C₂×C₁₃₁	C₂×C₃₂	1	1	yes	characteristic
17	C₁₃₁	M₁₂₈	1	1	yes	characteristic, complemented, Sylow
18	M₁₂₈	—	131	1	no	maximal, Sylow
19	C₂×C₃₂	D₁₃₁	1	1	yes	characteristic
20	C₆₄	—	131	2	no	—
21	C₂×C₁₆	Dic₁₃₁	1	1	yes	characteristic
22	C₃₂	Dic₁₃₁	1	1	yes	characteristic
23	C₃₂	C₂×D₁₃₁	1	1	yes	characteristic
24	C₂×C₈	C₁₃₁⋊C₈	1	1	yes	characteristic
25	C₁₆	C₁₃₁⋊C₈	1	1	yes	characteristic
26	C₁₆	C₂×Dic₁₃₁	1	1	yes	characteristic
27	C₂×C₄	C₁₃₁⋊C₁₆	1	1	yes	characteristic
28	C₈	C₁₃₁⋊C₁₆	1	1	yes	characteristic
29	C₈	C₂×C₁₃₁⋊C₈	1	1	yes	characteristic
30	C₂²	C₁₃₁⋊C₃₂	1	1	yes	characteristic
31	C₄	C₁₃₁⋊C₃₂	1	1	yes	characteristic
32	C₄	C₂×C₁₃₁⋊C₁₆	1	1	yes	characteristic
33	C₂	—	2	1	no	—
34	C₂	C₂×C₁₃₁⋊C₃₂	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃₂	C₂×D₁₃₁
commutator, socle	C₂×C₁₃₁	C₂×C₃₂
Fitting	C₂×C₃₂×C₁₃₁	C₂
prime radical	C₂²×C₁₃₁	C₃₂
cosocle	C₃₂×C₁₃₁	C₂²
hypercenter	C₂×C₃₂	D₁₃₁
hypocenter	C₁₃₁	M₁₂₈
LCS	G ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₃₂ ◁ C₂×C₃₂

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₁₃₁⋊C₆₄	C₂	semidirect
17. C₁₃₁	M₁₂₈	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₁₃₁⋊C₆₄	2
C₂×C₃₂×C₁₃₁	2
M₁₂₈	131